Correlated Energy Uncertainties in Reaction Rate Calculations
AAstronomy & Astrophysics manuscript no. CorrelatedEnergies c (cid:13)
ESO 2020July 9, 2020
Correlated Energy Uncertainties in Reaction Rate Calculations
Richard Longland , and Nicolas de S´er´eville North Carolina State University, Raleigh, NC 27695 Triangle Universities Nuclear Laboratory, Durham, NC 27708 Universit´e Paris-Saclay, CNRS / IN2P3, IJCLab, 91405 Orsay, FranceJuly 9, 2020
ABSTRACT
Context.
Monte Carlo methods can be used to evaluate the uncertainty of a reaction rate that arises from many uncertain nuclear in-puts. However, until now no attempt has been made to find the e ff ect of correlated energy uncertainties in input resonance parameters. Aims.
To investigate the impact of correlated energy uncertainties on reaction rates.
Methods.
Using a combination of numerical and Monte Carlo variation of resonance energies, the e ff ect of correlations are investi-gated. Five reactions are considered: two fictional, illustrative cases and three reactions whose rates are of current interest. Results.
The e ff ect of correlations in resonance energies depends on the specific reaction cross section and temperatures considered.When several resonances contribute equally to a reaction rate, and are located either side of the Gamow peak, correlations betweentheir energies dilute their e ff ect on reaction rate uncertainties. If they are both located above or below the maximum of the Gamowpeak, however, correlations between their resonance energies can increase the reaction rate uncertainties. This e ff ect can be hard topredict for complex reactions with wide and narrow resonances contributing to the reaction rate. Key words. methods: numerical methods: statistical nuclear reactions, nucleosynthesis, abundances
1. Introduction
Thermonuclear reaction rates dictate energy generation and ele-mental synthesis in stars and stellar explosions. They are a keyphysical input to computational stellar models that attempt toexplain astrophysical phenomena in conjunction with observa-tional data. It is for this reason that reaction rates must be wellknown, and moreover, their uncertainties must be well under-stood if comparisons between stellar models and observationsare to be made reliably. Reaction rate uncertainties were first ad-dressed by the Nuclear Astrophysics Compilation of REactionrates (NACRE) collaboration in an evaluation of 86 reactions onA =
2. Reaction Rate Formalism
What follows is a brief overview of astrophysical reaction rates.For more detail the reader is encouraged to refer to Rolfs &Rodney (1988); Angulo et al. (1999); Longland et al. (2010); a r X i v : . [ a s t r o - ph . I M ] J u l ichard Longland and Nicolas de S´er´eville: Correlated Energy Uncertainties in Reaction Rate Calculations Sallaska et al. (2013); Iliadis (2015). The reaction rate per parti-cle pair, (cid:104) σv (cid:105) , is defined as (cid:104) σv (cid:105) = (cid:32) πµ (cid:33) / kT ) / (cid:90) ∞ E σ ( E ) e − E / kT dE , (1)where µ is the reduced mass of the system, k is the Boltzmannconstant, T is the temperature at which the reaction rate is beingcalculated, and σ ( E ) is the energy-dependent cross section of thereaction.The cross section can be parameterised by removing the s-wave Coulomb barrier tunnelling cross section as σ ( E ) = E S ( E ) e − πη , (2)where η is the Sommerfeldt parameter defined by2 πη = . Z Z (cid:114) µ E . (3) Z and Z are the atomic numbers of the interacting particles.The quantity S ( E ) is the so-called Astrophysical S-Factor, whichcontains any details of the cross section that are not accountedfor by simple s-wave Coulomb barrier scattering. We see fromcombining Eqs. 2 and 1 that the reaction rate is defined as (cid:104) σv (cid:105) = (cid:32) πµ (cid:33) / kT ) / (cid:90) ∞ S ( E ) e − πη e − E / kT dE . (4)The product of the two exponentials: the Gamow factor, e − πη and the Boltzmann factor, e − E / kT approximates the energy rangeover which the astrophysical S-factor should be known. Thisproduct is the “Gamow peak”, and the location of the maximumis at E . This maximum and the width of the Gamow peak aredefined by: E = (cid:34)(cid:18) π (cid:126) (cid:19) ( Z Z e ) (cid:18) µ (cid:19) ( kT ) (cid:35) / = . (cid:16) Z Z µ T (cid:17) / (5) ∆ = √ (cid:112) E kT = . (cid:16) Z Z µ T (cid:17) / , (6)where Z and Z are the atomic numbers of the interacting nuclei, e is the elementary electric charge, and T is the temperature in10 K.Many reactions of astrophysical importance proceed throughnuclear resonances, populating compound nuclear states thatsubsequently decay. Note that we only consider 2-body reactionsin the following. For a single, isolated resonance, the cross sec-tion in Eq. (1) can be replaced by σ ( E ) = J + J + J + π k Γ a ( E ) Γ b ( E )( E − E r ) + Γ / . (7)Here, J , J , and J are the spins of the resonance, target, andprojectile particles, respectively. This angular momentum termis often denoted by the symbol, ω . Γ a ( E ) and Γ b ( E ) are energy-dependent quantities describing the entrance and exit partialwidths of the state in question. For example, for a (p, γ ) reaction, Γ a ( E ) corresponds to the proton partial width and Γ b ( E ) is the γ -ray partial width. Γ corresponds to the total width of the state,and E r is the resonance energy. For a charged particle, Γ a ( E ) isdefined as Γ a ( E ) = (cid:126) µ R P (cid:96) ( E ) C S θ a , (8) where P (cid:96) ( E ) is the energy-dependent penetration factor de-scribing the probability of the particles tunnelling though theCoulomb barrier, C S is the product of the Isospin Clebsh-Gordan coe ffi cient for the interacting particles and spectroscopicfactor. This latter quantity describes how well an excited statein the compound nucleus can be described by a single-particlestate. θ a is the dimensionless single-particle reduced width,which can be calculated theoretically from the particle’s wave-function (see Iliadis (1997)). We consider this latter quantityto be unity, here, since we’re only considering relative e ff ectsunder energy variations. The channel radius, R is calculated as R = R (cid:16) A / p + A / t (cid:17) . This choice does not have a large e ff ect onthe calculations provided it is used consistently throughout.If the resonance is su ffi ciently narrow such that its partialwidths do not vary significantly over its width, we can assumethat Γ a and Γ b are constant. In that case, the integral in Eqn. 1 canbe evaluated algebraically. Now the resonance cross section isreplaced by a single quantity: the resonance strength, ωγ , whichis defined by ωγ = ω Γ p Γ γ Γ p + Γ γ . (9)It’s important to recognise that the partial widths, Γ p and Γ γ ,in Eqn. 9 depend on resonance energies (through Eqn. 8 forcharged particles). Any energy shifts must be carefully prop-agated through these equations to determine shifted resonancestrengths. Once this procedure is followed, Eq. (1) can be re-placed with (cid:104) σv (cid:105) = (cid:32) πµ kT (cid:33) / (cid:126) (cid:88) i ωγ i e − E r / kT (10)This paper focuses on reaction rates dominated by resonances inthe absence of interference. Addressing interfering resonance re-action rates that are best described by R-matrix or other complexmodels is left for future work.
3. Monte Carlo Reaction Rates
In order to investigate the influence that correlated resonance en-ergy uncertainties have on reaction rates, the Monte Carlo reac-tion rate method first described in Longland et al. (2010) wasused as a starting point. The general strategy laid out in thatstudy was to first assign a probability density distribution to ev-ery uncertain input parameter to the reaction rate calculation. Forthe case of resonant cross sections, these uncertain parametersare the resonance energies, E r , partial widths, Γ a , and resonancestrengths, ωγ , in Eqns. 7, 9, and 10. The strategy was extendedto allow for uncertain spin-parities in Mohr et al. (2014).Once probability density distributions have been obtainedfor uncertain input parameters, sample parameters are randomlychosen from the distributions assuming that all parameters areindependent (the validity of this assumption is discussed below).The reaction rate calculated from the sampled parameters rep-resents a single sample rate. This procedure is repeated manytimes (10,000 is preferred but at least 3000 samples was foundto produce stable results) to obtain a distribution of reaction ratesthat can be summarised with a reaction rate probability densitydistribution. Longland et al. (2010) found that the reaction rateprobability density can often be summarised with a log-normaldistribution with shape parameters µ and σ . The recommended rate is then given by: (cid:104) σv (cid:105) rec. = e µ ( T ) . (11)The “low” and “high” rates given by the 1- σ uncertainties arefound using (cid:104) σv (cid:105) l = e µ ( T ) e − σ ( T ) (cid:104) σv (cid:105) h = e µ ( T ) e + σ ( T ) (12)The procedure described above was found to be a flexiblemethod for estimating the reaction rate uncertainties. Providedenergy uncertainties are correctly propagated into partial widthuncertainties, it works equally well for narrow and wide resonantreaction rates with large uncertainties. However, the method didnot account for the case in which there are correlations betweenparameters. Recently, e ff orts were made to include the e ff ects of correlatedpartial width and resonance strength uncertainties in the MonteCarlo method described above (Longland, 2017). This case ofcorrelated widths and resonance strengths needed to be investi-gated because resonances are usually normalised to some stan-dard, well known resonance. Given that experimental reality, theassumption of independent variables made in Longland et al.(2010) is not strictly accurate.Consider the following example: a partial width for reso-nance j , Γ j , is correlated with a reference resonance, r . Thesepartial widths carry factor uncertainties, f . u . j ≡ σ Γ , j / Γ j and f . u . r ≡ σ Γ , r / Γ r , respectively where σ Γ , j is the uncertainty in Γ j ,etc. A single correlation parameter, ρ j , can be used to describethe magnitude of their correlation: ρ j = σ Γ , r Γ r Γ j σ Γ , j ≡ f . u . r f . u . j . (13)During the Monte Carlo procedure, the following steps aretaken: First, random, uncorrelated samples are produced for thereference resonance and resonance j . These samples, denoted x i and y j , i , are drawn from standard normal distributions (i.e., witha mean of zero and standard deviation of 1). Longland (2017)showed that a simple, 2-step procedure can then be used to cal-culate correlated partial widths. Second, the samples are corre-lated using y (cid:48) j , i = ρ j x i + (cid:113) − ρ j y j , i , (14)where y (cid:48) j , i are the correlated samples for resonance j . Finally, thepartial width samples can be computed with the knowledge thatpartial widths should be log-normally distributed (see Longlandet al. (2010)): Γ j , i = Γ j , rec. ( f . u . ) y (cid:48) j , i . (15)Here, the recommended partial width for resonance j is Γ j , rec. .Note that the probability density distributions defined byEq. 15 are a log-normal distribution. For small uncertainties theyresemble Gaussian distributions, but are not defined for negativevalues. They therefore describe physical parameters such as par-tial widths and resonance strengths appropriately. They are alsowell motivated by the central limit theorem, as discussed in de-tail in Longland et al. (2010).In the present study we follow the same strategy as above,but must consider two modifications: (i) energy uncertaintiesmust be propagated through partial widths. For the charged par-ticle reactions considered here, this is accomplished using Eq. 8; and (ii) energy uncertainties are not log-normal. Often large res-onance energy uncertainties arise because of large uncertaintiesin reaction Q-values. For example, imagine a low-energy reso-nance with a large energy uncertainty that has an appreciableprobability of being a sub-threshold resonance. Modification (i)is already accounted for in the RatesMC code implementation ofthe Monte Carlo methods outlined in Longland et al. (2010) The second case requires modifications to Eqns. 13 and 15: ρ j = min( σ E ) σ E , j . (16) E j , i = E j , rec. + y (cid:48) j , i σ E , j . (17)The correlated standard normal samples, y (cid:48) j , i are computed us-ing Eq. 14. The purpose of Eq. 16 is similar to that of Eq. 13. Itensures that correlations between resonance energies do not ex-ceed their experimental limits. For example, consider a reactionin which the majority of the resonance energy uncertainty arisesfrom an uncertain reaction Q-value, σ Q . Those resonance ener-gies will be correlated with ρ ≈
1. Now assume another reso-nance in that reaction populates an excited state that also has anuncertain excitation energy, σ Ex . The energy of that resonancewill have a larger overall uncertainty (given by σ = σ Q + σ Ex )and will not be fully correlated with the other resonance ener-gies. For this example resonance, ρ < ff erent means must be considered. In most cases, highenergy resonance energies are determined through direct mea-surement of the reaction cross section. Resonance energies canbe determined by measuring a yield curve, for example. Low-energy resonances in the same reaction may be determined fromexcitation energies and an uncertain Q-value. Therefore, the abil-ity to enable or disable energy correlations for a resonance mustbe available. The RatesMC code has been modified to allow this.
4. Test Cases
To investigate the e ff ect of correlated energy uncertainties onreaction rates, two fictional reactions designed to probe low- andhigh-density resonance regimes are first investigated. The e ff ecton actual reaction rates will be detailed in Sec. 5. The first test case considered is shown in Fig. 1. The physicalsystem consists of a fictional reaction whose cross section con-sists entirely of three isolated, narrow resonances. Thus, Eqn. 10can be used to calculate the reaction rate. The partial widths arecalculated using Eqn. 8 by assuming R = .
25 fm and a (p, γ )reaction occurring on an isotope with an atomic number of 12and mass of 23. The resonance parameters chosen are shown inTab. 1.These resonances contribute di ff erent amounts to the reac-tion rate depending on the temperature. The low-energy charged-particle reaction resonances most important at low temperaturescan be well predicted by the Gamow peak defined in Eqn. 4.Indeed, this is the case as shown in Fig. 1, where the threeresonance contributions to the total reaction rate follow closelythe progression of the Gamow peak as temperatures increase. Note that the correlation between resonance energy and partialwidths makes reaction rate uncertainty propagation using calculatedresonance strengths unreliable. Partial widths should be used wheneveravailable. 3ichard Longland and Nicolas de S´er´eville: Correlated Energy Uncertainties in Reaction Rate Calculations a) b) c) E (MeV)
Fig. 1. (colour online) Test cases for our investigation of the impact of energy uncertainty correlations on nuclear reaction rateuncertainties. Shown as a solid line is the Gamow peak in arbitrary units calculated from Eqn. 10 at three temperatures: 0.08 GK,0.13 GK, and 0.2 GK in panels a), b), and (c) respectively. The reaction rate for each of the three narrow resonances described inTab. 1 are shown as coloured points. The y-direction is scaled arbitrarily for clarity to highlight which resonances contribute most tothe reaction rate and where they are located in comparison to the Gamow peak. This information is also displayed by the bar on theright of each panel. For example, in panel a), only one resonance – the E c.m. r =
100 keV resonance shown in green – contributes tothe reaction rate. In panel b), two resonances – E c.m. r =
100 keV and E c.m. r =
250 keV in green and orange – contribute approximately50% each to the reaction rate at 0.13 GK. E c.m. r (keV) J π C S Γ p (eV) Γ γ (eV) ωγ (eV)100.0 1 − . × − . × − − . × − . × − − . × . × − Table 1.
Parameters of the three resonances considered in ourfictional test case. E c.m. r is the centre-of-mass resonance energy(in keV).Higher energy resonances, though, will not follow this pat-tern. Their cross sections become constrained by γ -ray partialwidths, which does not exhibit the Coulomb barrier energy de-pendence (Newton et al., 2007).First, consider the reaction rate at T = .
13 GK. At this tem-perature, the two resonances at E c.m. r =
100 keV and E c.m. r =
250 keV contribute approximately equally to the total reactionrate as shown by the bar on the right of the centre panel. Theyare situated either side of the maximum of the Gamow peak. Toinvestigate the e ff ect that their energy uncertainties have on thereaction rate, their resonance energies are varied over a givenrange. For each trial resonance energy, the proton partial widthis re-calculated using Eqn. 8 and the parameters in Tab. 1. Usingthis, the reaction rate is determined using Eqs. 9 and 10. Theseresonance energies are varied using two schemes: (i) correlatedenergies, so any increase in the energy of one resonance corre-sponds to an equal increase in the other, and (ii) anti-correlatedenergies, in which any energy increase in one resonance energycorresponds to an equal magnitude decrease in the other. Theresonance energies are varied by ±
60 keV this way. The varia-tions a ff ect the individual contributions to the rate as well as thetotal rate. These are shown in Fig. 2.Figure 2 shows that larger reaction rate variations are ex-pected if resonance energies are anti-correlated. To understandthis e ff ect, consider first the correlated energy case in the left-hand panel as well as the middle panel of Fig. 1. As the res- onances both increase in energy, the one at E c.m. r =
100 keVshifts closer to the maximum of the Gamow peak, thus increas-ing its contribution to the total rate. Conversely, the resonance at E c.m. r =
250 keV moves away from the maximum of the Gamowpeak and decreases its contribution. The net e ff ect is that the totalrate does increase, but the magnitude is weakened by the oppo-site contributions of the two resonances. As the resonance ener-gies decrease, a similar e ff ect is apparent: the E c.m. r =
100 keVresonance contributes less while the E c.m. r =
250 keV resonancecontributes more, resulting again in an increase in reaction ratethat is weakened by the opposite contributions.In the case of anti-correlated resonance energies in the right-hand panel of Fig. 2, the resonances contributions work in tan-dem. When the E c.m. r =
100 keV resonance energy is decreased,it moves away from the maximum of the Gamow peak to lowerenergies, while the E c.m. r =
250 keV resonance also moves away,but to higher energies. Thus the contributions of both resonancesdecrease, resulting in a reduced total reaction rate. Similarly, asone moves towards the maximum of the Gamow peak, so willthe other, resulting in a strengthened increase in the total rate.Anti-correlated energy uncertainties in this case result in an in-creased reaction rate uncertainty.The anti-correlated resonance energy example discussedabove will rarely occur in experimental resonance energy mea-surements. However, if the resonance energies are treated ascompletely uncorrelated during the Monte Carlo procedure, theirrelative variations will be somewhere between the fully corre-lated and fully anti-correlated cases. Thus, taking into accountthe e ff ect illustrated in Fig. 2 we expect larger uncertainties foruncorrelated resonance energies than correlated energies. Thisis indeed the case, as shown in Fig. 3, which shows the reac-tion rate probability density distributions for these two cases. Ingrey, a broad, approximately Gaussian peak represents the prob-ability density distribution for the uncorrelated energy case. Inthe correlated case, the e ff ect shown in Fig. 2 is clear in that theprobability distribution is not only narrower, but is also highly Energy Shift (keV) R a t i o −7 −5 −3 −1 −60 −40 −20 0 20 40 60 Correlated Energies
Energy Shift (keV) R a t i o −7 −5 −3 −1 −60 −40 −20 0 20 40 60 Anti−correlated Energies
Fig. 2. (colour online) The e ff ects of correlated and anti-correlated energy variations to a pair of resonances on the calculatedreaction rate at 0.13 GK. The green and orange dashed lines show the individual resonance contributions from the E c.m. r =
100 keVand E c.m. r =
250 keV resonances to the total reaction rate shown by the black line. The Energy shift is defined as the shift of thelower energy resonance. All are normalised to their recommended values at ∆ E = only increase astheir energies are varied. Factor Difference P r obab ili t y D en s i t y Fig. 3. (colour online) Reaction rate probability distributionsfrom the Monte Carlo variation of resonance energies at 0.13GK. Shown in grey is the uncorrelated case, in which the reso-nance energies are allowed to vary independently. In blue is thecase in which the resonance energies are fully correlated. Thedistribution becomes narrower and highly skewed in this caseowing to the e ff ect illustrated in Fig. 2.How universal is this e ff ect? In the example above, a specifictemperature was chosen to correspond to two resonances eitherside of the Gamow peak. To investigate more possibilities, thethird panel in Fig. 1 is illustrative. In this case, the resonances at E c.m. r =
250 keV and E c.m. r =
300 keV contribute about 75% and 25% to the reaction rate, respectively. They’re also both locatedabove the maximum of the Gamow peak, so as we shift their en-ergies in a correlated manner, they will both shift toward or awayfrom the peak in unison. The e ff ect of their variations on the totalreaction rate is shown in Fig. 4. In this case, the opposite e ff ectto that observed in Fig. 2 is apparent. If the resonances are corre-lated, they move together to reduce or increase the reaction rate.If they are anti-correlated, their contributions essentially cancelout to produce very little variation in the total reaction rate.The Monte Carlo reaction rate comparison at T = . ff ect to theexample at T = .
13 GK. If energy correlations are taken intoaccount, the reaction rate uncertainties increase . Clearly, thesee ff ects are hard to predict, particularly when large resonance en-ergies are concerned. However, using Monte Carlo uncertaintypropagation, we are able to account for the e ff ect of energy cor-relations on the reaction rate uncertainties. Following the same procedure for a high density of resonancesproduces results that are easier to predict. In this case, the re-action rate uncertainty when correlated energy uncertainties aretaken into account reliably decreases . Now, resonance are placedat energies between E c.m. r =
50 keV and E c.m. r =
400 keV with aspacing of 20 keV.The reaction rate uncertainty is shown in Fig. 6 for T = .
13 GK. The uncertainty for correlated energies (blue) clearlydecreases in comparison to the uncorrelated case (grey). As tem-perature increases we find that the e ff ect of correlations de-creases because more resonances contribute to the rate. The results discussed above are di ffi cult to predict a priori.However, a conservative estimate of reaction rate uncertaintieswould only be concerned with the case in which resonance en-ergy correlations increase the reaction rate uncertainties. Thiscase is shown in Fig. 5. For these cases to occur, the resonances Energy Shift (keV) R a t i o -60 -40 -20 0 20 40 60 Correlated Energies
Energy Shift (keV) R a t i o -60 -40 -20 0 20 40 60 Anti-correlated Energies
Fig. 4. (colour online) The e ff ects of correlated and anti-correlated resonance energies on the reaction rate at T = . E c.m. r =
250 keV and E c.m. r =
300 keV resonances, respectively. SeeFig. 2 for details.
Factor Difference P r obab ili t y D en s i t y Fig. 5. (colour online) Reaction rate probability distributionsfrom the Monte Carlo variation of resonance energies at 0.2GK. See Fig. 3 for details. The probability distribution becomeswider when correlated energies are considered in this case owingto the e ff ect illustrated in Fig. 4.should be located on the same side of the Gamow peak (i.e., astheir energy increases / decreases, correlations would cause themall to increase / decrease their contributions to the total rate in uni-son). This requires more than one resonance contributing to therate, but the resonance density cannot be too high, else reso-nances would be distributed throughout the Gamow peak. Forexample, it would not be realistic to expect a case where a highresonance density is found on one side of the Gamow peak, butno resonances on the other. Furthermore, the resonance energyuncertainty in contributing resonances should be large in com-parison to the Gamow peak’s width, defined by Eq. (6). Factor Difference P r obab ili t y D en s i t y Fig. 6. (colour online) Reaction rate probability distributionsfrom the Monte Carlo variation of resonance energies at 0.13GK for the high resonance density case. See Fig. 3 for details.
5. Physical Cases
Now that the general behaviour of how reaction rate uncer-tainties change when resonance energy correlations are takeninto account, some physically realistic cases will be considered.These are the Ar(p, γ ) K reaction; the Ca(p, γ ) Sc reaction;and the N( α ,p) O reaction. They span a range of resonancedensities and represent cases where the resonance energies areknown to be uncertain and correlated. Ar(p, γ ) K Reaction
The Ar(p, γ ) K reaction is a key reaction in explosive hydro-gen burning. In x-ray bursts this reaction is expected to occurfaster than its competing β -decay, but in novae (i.e., lower tem-peratures) the rate is less well known. Its e ff ect on the nucle- osynthesis of heavier elements is not well understood (Glasner& Truran, 2009). The reaction rate was evaluated in Iliadis et al.(1999). At that time, the Q-value of this reaction was poorlyknown (Audi & Wapstra, 1995), leading to large, correlated reso-nance energy uncertainties. Since that time the excitation energyuncertainties in K have been dramatically reduced by an orderof magnitude by Wrede et al. (2010). However, the resonanceenergies are still expected to be correlated and the resonancedensity of this reaction is very low with just 4 known resonancesbelow 1 MeV. For these reasons it is an ideal case with whichto investigate correlated resonance energies in the Monte Carloframework. The resonance parameters from Wrede et al. (2010)are listed in Tab. 2. Note that we have assigned very small uncer-tainties (1%) to the partial widths so that the resonance energye ff ects can be clearly identified. Separate calculations confirmthat the uncertainties due to other sources sum quadratically, asexpected. E c.m. r (keV) J π Γ p (eV) Γ γ (eV)48.4 (8) 2 − . × − . × − + . × − . × − − . × − . × − + . × . × − Table 2.
Resonance parameters for the Ar(p, γ ) K reactiontaken from Wrede et al. (2010). The uncertainties in Γ p and Γ γ have been assumed to be 1% to emphasise the e ff ect of the en-ergy uncertainties (see text).The reaction rate uncertainties for the Ar(p, γ ) K reactionassuming the resonance parameters shown in Tab. 2 are shown inFig. 7. The coloured contour represents the reaction rate uncer-tainties arising from correlated energy uncertainties, with thickand thin lines representing the 1 σ and 2 σ uncertainty bands,respectively. These rates have been normalised to the recom-mended (median) rate, which is shown by a horizontal line atunity. The blue lines show the reaction rate uncertainties whenresonance energy correlations are not taken into account. Thethick blue line represents the median rate, and the thin dashedblue lines represent the 1 σ uncertainties. This figure shows thatover most of the temperature range, energy correlations do notstrongly a ff ect the reaction rate uncertainties. They are onlyslightly smaller when taking resonance energy correlations intoaccount. This is mostly due to the fact that below 200 MK andabove 1 GK, only one resonance is contributing and the e ff ect ofenergy correlation is almost in-existent. In between these tem-peratures two resonances contribute to the reaction rate. At 400MK, for example, the resonances are located either side of themaximum of the Gamow peak. In this case the e ff ect of corre-lations is small, but in line with the case described in Sec. 4.1:as the E c.m. r =
260 keV resonance moves to a lower energy, forexample, the E c.m. r =
623 keV resonance also moves to a lowerenergy, thus reducing the impact of resonance energies on thereaction rate.These calculations were also performed assuming the (ob-solete) resonance parameters reported in Iliadis et al. (1999).Additionally the total rate uncertainty is much larger owing tothe larger energy uncertainties, the correlations between thoseenergies have the same, minor, e ff ect on the uncertainty as out-lined above. Fig. 7. (colour online) Reaction rate uncertainties for the Ar(p, γ ) K reaction assuming the resonance parameters shownin Tab. 2. Recall that only resonance energies are taken into ac-count. The rate has been normalised to the median rate, whichis shown by a dashed line at unity. The thick and thin blacklines represent the 1 σ and 2 σ uncertainties in the correlated res-onance energy calculation. The blue lines show the uncorrelatedcase, with the thick line representing the median rate (again, nor-malised to the recommended correlated energy rate), and dashedlines showing the 1 σ uncertainties. At T =
400 MK, the e ff ectof correlations slightly decreases the reaction rate uncertainty. Ca(p, γ ) Sc Reaction
The Ca(p, γ ) Sc reaction is also important in explosive nucle-osynthesis. It influences the end-point of the rp-process in x-raybursts, and has also been evaluated in Iliadis et al. (1999). Theresonance density is similar to the Ar(p, γ ) K reaction, but inthis case the Q-value is better known. The resonance energy un-certainties are just 5-6 keV, as shown in Tab. 3. E c.m. r (keV) J π Γ p (eV) Γ γ (eV)223 (5) 2 − . × − . × −
353 (5) 5 − . × − . × − − . × . × − − . × . × − Table 3.
Resonance parameters for the Ca(p, γ ) Sc reactiontaken from Iliadis et al. (1999). Note that the two resonances at E c.m. r = Ca( He,t) Sc measurement by Schulz et al. (1971). Theuncertainties in Γ p and Γ γ have been assumed to be 1% to em-phasise the e ff ect of the energy uncertainties (see text).In this case, the predictions in Sec. 4.3 indicate that thereshould, indeed, be an e ff ect of resonance energy correlations onthe reaction rate. The average resonance energy separation is 300keV compared with ∆ =
200 keV at 400 MK. In contrast to the Ar(p, γ ) K reaction, though, there is a temperature range atwhich both resonances at E c.m. r =
223 keV and E c.m. r =
353 keVlay on the low-energy side of the Gamow peak. Thus we expectan e ff ect of correlated energies on the reaction rate.Figure 8 shows that correlations between resonance en-ergy uncertainties do, indeed, a ff ect the reaction rate uncer-tainty strongly at 300-400 MK where both resonances at E c.m. r =
233 keV and E c.m. r =
353 keV contribute to the reaction rate andare both are on the low-energy side of the Gamow peak. In thisparticular scenario, as the resonance co-move to lower energies, both contribute to a lower reaction rate. Conversely if they bothco-move to higher resonance energy they both contribute to ahigher reaction rate. In the uncorrelated energy case, this sce-nario is more rare, thus the rate uncertainty is smaller. Note thatover most of the temperature range, 0 . < T <
1, the rate un-certainties are considerable. At those temperatures, only 1 or 2resonances ever contributes towards the rate, and the uncertaintyis arising from the strong Coulomb barrier energy dependence inEq. (8).
Fig. 8. (colour online) Reaction rate uncertainties for the Ca(p, γ ) Sc reaction assuming the resonance parametersshown in Tab. 3 and accounting only for uncertainties in the res-onance energies. See Fig. 7 for description. N( α ,p) O Reaction
The N( α ,p) O reaction a ff ects nitrogen production in super-nova explosions as the shock-wave passes through the outer re-gions of the exploding star (Pignatari et al., 2013). That mate-rial can eventually go on to form pre-solar grains, whose iso-topic nitrogen ratios provide a precise test of astrophysical mod-els (Zinner, 2014). The N( α ,p) O reaction rate should beknown, therefore, to a high precision. The rate was recentlyevaluated in Meyer et al. (2020). In this case the N + α thresh-old ( S α + N = . F su ff ers from systematic uncertainty of sev- eral tens of keV which introduces a strong correlation betweenresonance energies. This uncertainty originates from a possibleerror in the calibration of one of the magnets used during themeasurement of the excitation functions of the O(p,p) O re-action by Salisbury et al. (1962); Salisbury & Richards (1962),and the O(p,p’) O and O(p, α ) N reactions by Dangle et al.(1964). The properties of the most influential resonances with α -particle partial width determined either from direct measure-ments or mirror symmetry considerations are taken from Meyeret al. (2020) and summarised in Tab. 4. E c.m. r (keV) J π Γ α (keV) Γ p (keV)741 (20) 1 / + × − / − × − / + / − × − / +
11 1352255 (30) 5 / +
14 792405 (40) 3 / −
25 636
Table 4.
Resonance parameters for the N( α ,p) O reactiontaken from Meyer et al. (2020). The uncertainties in Γ p and Γ γ have been assumed to be 1% to emphasise the e ff ect of the en-ergy uncertainties (see text)In order to emphasise the e ff ect of the (un)correlated uncer-tainties on the resonance energy a very small uncertainty (1%)has been assigned to the partial widths and the tentative spinand parity have been considered as firmed assignment. In thisparticular case, the resonances have very large (factor of 2.5) un-certainties, which completely dominates the energy uncertaintiesunder investigation here.Even though some resonances have large total widths theirnumber is relatively small and they can be considered as isolated.The case of low resonance density discussed in Sec. 4.1 shouldthen apply and an e ff ect of correlated energies on the reactionrate is then expected. This is indeed the case as shown in Fig. 9where the N( α ,p) O reaction rate uncertainties are presented.Let’s first consider a typical temperature of 1 GK which cor-responds to a Gamow peak with a maximum of about 1 MeVand width of ∆ ≈
700 keV from Eq. 6. In this case the reactionrate is dominated by the two resonances at E c.m. r =
741 keV and E c.m. r = E c.m. r =
741 keVresonance dominates the rate and is located on the low-energy
Fig. 9. (colour online) Reaction rate uncertainties for the N( α ,p) O reaction assuming the resonance parameters shownin Tab. 4 and accounting only for uncertainties in the resonanceenergies. See Fig. 7 for description.tail of the Gamow peak. Any variation in the resonance energyhas a large e ff ect on the rate through Eq. (8).
6. Summary and Conclusions
Monte Carlo methods can be a powerful tool for computing sta-tistically rigorous uncertainties of thermonuclear reaction rates.While the methods have been in use for some time, no e ff orthad previously been made to account for correlations betweenresonance energies. These e ff ects are particularly important forradioactive nuclei where resonances are not often directly mea-sured.In this paper, we expanded on the correlation scheme de-veloped in Longland (2017) to allow for correlations betweenresonance energies. We found that the e ff ects are not necessarilyeasy to predict. Reactions rates dominated by many resonancesare not strongly a ff ected by correlations, whereas those domi-nated by only a single resonance at astrophysically importanttemperatures are also not significantly a ff ected. The cases thatmatter most are those where multiple resonances contributed tothe reaction rate. This e ff ect is enhanced if they are on the sameside of the Gamow peak’s maximum value.Correlations between resonance parameters can be an impor-tant e ff ect in thermonuclear reaction rate calculations. The corre-lation of resonance energies was previously unexplored, whichhas now been accounted for in this work. Since the e ff ects ofthese correlations are rather unpredictable, we recommend thatany reaction rate uncertainty calculation be carefully checked toensure corrections due to resonance energy correlations do notsignificantly a ff ect the results. This will be of particular impor-tance for reactions on isotopes far from stability, where the en-ergies of excited states can carry large, correlated uncertaintiesbecause they are determined from uncertain reaction Q-values. Acknowledgements.
This material is based partly upon work supported by theU.S. Department of Energy, O ffi ce of Science, O ffi ce of Nuclear Physics, underAward Number de-sc0017799. References